 Scalar-gradient inclination in a steady planar strain

For horizontal strain matrix $S=\left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)$ and zero vertical velocity, horizontal scalar gradients obey ${\dot{k}}_h=-S{k}_h$. Since $S^2=c^{2}I$ with $c=\sqrt{a^2+b^2}$, the growing projection is $(I-S/c)k_h(0)/2$. If nonzero it equals $k_0(\cos \varphi ,\sin \varphi )$ and grows as $e^{ct}$. Fixed vertical shears give ${\dot{k}}_3=-{\sigma }_{13}{k}_1-{\sigma }_{23}{k}_2$, so $k_3\sim -k_{0}D{e}^{ct}/c$ with $D={\sigma }_{13}\cos \varphi +{\sigma }_{23}\sin \varphi$. The ratio $\alpha =|k_3|/|k_h|$ tends to the displayed value. It measures scalar-isosurface orientation; zero growing projection or $D=0$ must not be described as generic exponential vertical-gradient growth.